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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.cubic_hermite"></a><a class="link" href="cubic_hermite.html" title="Cubic Hermite interpolation">Cubic Hermite interpolation</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.cubic_hermite.h0"></a>
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<span class="phrase"><a name="math_toolkit.cubic_hermite.synopsis"></a></span><a class="link" href="cubic_hermite.html#math_toolkit.cubic_hermite.synopsis">Synopsis</a>
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</h4>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">interpolators</span><span class="special">/</span><span class="identifier">cubic_hermite</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span> <span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cubic_hermite</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
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<span class="identifier">cubic_hermite</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">abscissas</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">ordinates</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span><span class="special">&&</span> <span class="identifier">derivatives</span><span class="special">);</span>
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<span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">void</span> <span class="identifier">push_back</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dydx</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">friend</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="keyword">operator</span><span class="special"><<(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span> <span class="special">&</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">cubic_hermite</span> <span class="special">&</span> <span class="identifier">m</span><span class="special">);</span>
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<span class="special">};</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cardinal_cubic_hermite</span> <span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
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<span class="identifier">cardinal_cubic_hermite</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">y</span><span class="special">,</span> <span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">dydx</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dx</span><span class="special">)</span>
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<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">cardinal_cubic_hermite_aos</span> <span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">using</span> <span class="identifier">Point</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">Point</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
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<span class="identifier">cardinal_cubic_hermite_aos</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&&</span> <span class="identifier">data</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">dx</span><span class="special">);</span>
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<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="keyword">inline</span> <span class="identifier">Real</span> <span class="identifier">prime</span><span class="special">(</span><span class="identifier">Real</span> <span class="identifier">x</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">domain</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="special">}</span> <span class="comment">// namespaces</span>
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</pre>
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<h4>
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<a name="math_toolkit.cubic_hermite.h1"></a>
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<span class="phrase"><a name="math_toolkit.cubic_hermite.cubic_hermite_interpolation"></a></span><a class="link" href="cubic_hermite.html#math_toolkit.cubic_hermite.cubic_hermite_interpolation">Cubic Hermite
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Interpolation</a>
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</h4>
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<p>
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The cubic Hermite interpolant takes non-equispaced data and interpolates between
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them via cubic Hermite polynomials whose slopes must be provided. This is a
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very nice interpolant for solution skeletons of ODEs steppers, since numerically
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solving <span class="emphasis"><em>y</em></span>' = <span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>,
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<span class="emphasis"><em>y</em></span>) produces a list of positions, values, and their derivatives,
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i.e. (<span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>y</em></span>,<span class="emphasis"><em>y</em></span>')=(<span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>y</em></span>,<span class="emphasis"><em>f</em></span>(<span class="emphasis"><em>x</em></span>,<span class="emphasis"><em>y</em></span>))-which
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is exactly what is required for the cubic Hermite interpolator. The interpolant
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is <span class="emphasis"><em>C</em></span><sup>1</sup> and evaluation has 𝑶(log(<span class="emphasis"><em>N</em></span>))
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complexity. An example usage is as follows:
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</p>
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<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span> <span class="number">5</span><span class="special">,</span> <span class="number">9</span> <span class="special">,</span> <span class="number">12</span><span class="special">};</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">y</span><span class="special">{</span><span class="number">8</span><span class="special">,</span><span class="number">17</span><span class="special">,</span> <span class="number">4</span><span class="special">,</span> <span class="special">-</span><span class="number">3</span><span class="special">};</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span> <span class="identifier">dydx</span><span class="special">{</span><span class="number">5</span><span class="special">,</span> <span class="special">-</span><span class="number">2</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">};</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cubic_hermite</span><span class="special">;</span>
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<span class="keyword">auto</span> <span class="identifier">spline</span> <span class="special">=</span> <span class="identifier">cubic_hermite</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">dydx</span><span class="special">));</span>
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<span class="comment">// evaluate at a point:</span>
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<span class="keyword">double</span> <span class="identifier">z</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span>
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<span class="comment">// evaluate derivative at a point:</span>
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<span class="keyword">double</span> <span class="identifier">zprime</span> <span class="special">=</span> <span class="identifier">spline</span><span class="special">.</span><span class="identifier">prime</span><span class="special">(</span><span class="number">3.4</span><span class="special">);</span>
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</pre>
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<p>
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Sanity checking of the interpolator can be achieved via:
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</p>
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<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">spline</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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</pre>
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<p>
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Note that the interpolator is pimpl'd, so that copying the class is cheap,
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and hence it can be shared between threads. (The call operator and <code class="computeroutput"><span class="special">.</span><span class="identifier">prime</span><span class="special">()</span></code>
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are threadsafe; <code class="computeroutput"><span class="identifier">push_back</span></code> is
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not.)
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</p>
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<p>
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This interpolant can be updated in constant time. Hence we can use <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span></code>
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to do real-time interpolation:
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</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">circular_buffer</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="special">...</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_x</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">};</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_y</span><span class="special">{</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">};</span>
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<span class="identifier">boost</span><span class="special">::</span><span class="identifier">circular_buffer</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">initial_dydx</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
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<span class="keyword">auto</span> <span class="identifier">circular_hermite</span> <span class="special">=</span> <span class="identifier">cubic_hermite</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">initial_x</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">initial_y</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">initial_dydx</span><span class="special">));</span>
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<span class="comment">// interpolate via call operation:</span>
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<span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_hermite</span><span class="special">(</span><span class="number">3.5</span><span class="special">);</span>
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<span class="comment">// add new data:</span>
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<span class="identifier">circular_hermite</span><span class="special">.</span><span class="identifier">push_back</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">8</span><span class="special">);</span>
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<span class="comment">// interpolate at 4.5:</span>
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<span class="identifier">y</span> <span class="special">=</span> <span class="identifier">circular_hermite</span><span class="special">(</span><span class="number">4.5</span><span class="special">);</span>
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</pre>
|
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<p>
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For the equispaced case, we can either use <code class="computeroutput"><span class="identifier">cardinal_cubic_hermite</span></code>,
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which accepts two separate arrays of <code class="computeroutput"><span class="identifier">y</span></code>
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and <code class="computeroutput"><span class="identifier">dydx</span></code>, or we can use <code class="computeroutput"><span class="identifier">cardinal_cubic_hermite_aos</span></code>, which takes a
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vector of <code class="computeroutput"><span class="special">(</span><span class="identifier">y</span><span class="special">,</span> <span class="identifier">dydx</span><span class="special">)</span></code>,
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i.e., and array of structs (<code class="computeroutput"><span class="identifier">aos</span></code>).
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||
The array of structs should be preferred as it uses cache more effectively.
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||
</p>
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<p>
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||
Usage is as follows:
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||
</p>
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||
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cardinal_cubic_hermite</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">x0</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
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||
<span class="keyword">double</span> <span class="identifier">dx</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">y</span><span class="special">(</span><span class="number">128</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">dydx</span><span class="special">(</span><span class="number">128</span><span class="special">,</span> <span class="number">0</span><span class="special">);</span>
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<span class="keyword">auto</span> <span class="identifier">ch</span> <span class="special">=</span> <span class="identifier">cardinal_cubic_hermite</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">y</span><span class="special">),</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">dydx</span><span class="special">),</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">dx</span><span class="special">);</span>
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||
</pre>
|
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<p>
|
||
For the "array of structs" version:
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||
</p>
|
||
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">interpolators</span><span class="special">::</span><span class="identifier">cardinal_cubic_hermite_aos</span><span class="special">;</span>
|
||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">>></span> <span class="identifier">data</span><span class="special">(</span><span class="number">128</span><span class="special">);</span>
|
||
<span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&</span> <span class="identifier">datum</span> <span class="special">:</span> <span class="identifier">data</span><span class="special">)</span> <span class="special">{</span>
|
||
<span class="identifier">datum</span><span class="special">[</span><span class="number">0</span><span class="special">]</span> <span class="special">=</span> <span class="number">1</span><span class="special">;</span> <span class="comment">// y</span>
|
||
<span class="identifier">datum</span><span class="special">[</span><span class="number">1</span><span class="special">]</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="comment">// y'</span>
|
||
<span class="special">}</span>
|
||
|
||
<span class="keyword">auto</span> <span class="identifier">ch</span> <span class="special">=</span> <span class="identifier">cardinal_cubic_hermite_aos</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">data</span><span class="special">),</span> <span class="identifier">x0</span><span class="special">,</span> <span class="identifier">dx</span><span class="special">);</span>
|
||
</pre>
|
||
<p>
|
||
For a quick sanity check, we can call <code class="computeroutput"><span class="identifier">ch</span><span class="special">.</span><span class="identifier">domain</span><span class="special">()</span></code>:
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">auto</span> <span class="special">[</span><span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">x_max</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">ch</span><span class="special">.</span><span class="identifier">domain</span><span class="special">();</span>
|
||
</pre>
|
||
<h4>
|
||
<a name="math_toolkit.cubic_hermite.h2"></a>
|
||
<span class="phrase"><a name="math_toolkit.cubic_hermite.performance"></a></span><a class="link" href="cubic_hermite.html#math_toolkit.cubic_hermite.performance">Performance</a>
|
||
</h4>
|
||
<p>
|
||
Google benchmark was used to evaluate the performance.
|
||
</p>
|
||
<pre class="programlisting"><span class="identifier">Run</span> <span class="identifier">on</span> <span class="special">(</span><span class="number">16</span> <span class="identifier">X</span> <span class="number">4300</span> <span class="identifier">MHz</span> <span class="identifier">CPU</span> <span class="identifier">s</span><span class="special">)</span>
|
||
<span class="identifier">CPU</span> <span class="identifier">Caches</span><span class="special">:</span>
|
||
<span class="identifier">L1</span> <span class="identifier">Data</span> <span class="number">32</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
|
||
<span class="identifier">L1</span> <span class="identifier">Instruction</span> <span class="number">32</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
|
||
<span class="identifier">L2</span> <span class="identifier">Unified</span> <span class="number">1024</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x8</span><span class="special">)</span>
|
||
<span class="identifier">L3</span> <span class="identifier">Unified</span> <span class="number">11264</span> <span class="identifier">KiB</span> <span class="special">(</span><span class="identifier">x1</span><span class="special">)</span>
|
||
<span class="identifier">Load</span> <span class="identifier">Average</span><span class="special">:</span> <span class="number">1.16</span><span class="special">,</span> <span class="number">1.11</span><span class="special">,</span> <span class="number">0.99</span>
|
||
<span class="special">-----------------------------------------------------</span>
|
||
<span class="identifier">Benchmark</span> <span class="identifier">Time</span>
|
||
<span class="special">-----------------------------------------------------</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">9.67</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">10.4</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">10.9</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">12.3</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">13.4</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">14.9</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">15.5</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">18.0</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">19.9</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">23.0</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">25.3</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">28.9</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">31.0</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">1.32</span> <span class="identifier">lgN</span>
|
||
<span class="identifier">CubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">13</span> <span class="special">%</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">3.14</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">3.13</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">3.19</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">3.14</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">3.38</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">3.54</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">3.51</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">3.76</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">5.82</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">5.76</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">5.85</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">5.69</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">5.77</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">4.28</span> <span class="special">(</span><span class="number">1</span><span class="special">)</span>
|
||
<span class="identifier">CardinalCubicHermite</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">28</span> <span class="special">%</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">256</span> <span class="number">3.22</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">512</span> <span class="number">3.22</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1024</span> <span class="number">3.26</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">2048</span> <span class="number">3.23</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">4096</span> <span class="number">3.49</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">8192</span> <span class="number">3.57</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">16384</span> <span class="number">3.73</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">32768</span> <span class="number">4.12</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">65536</span> <span class="number">4.13</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">131072</span> <span class="number">4.12</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">262144</span> <span class="number">4.12</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">524288</span> <span class="number">4.12</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">>/</span><span class="number">1048576</span> <span class="number">4.19</span> <span class="identifier">ns</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_BigO</span> <span class="number">3.73</span> <span class="special">(</span><span class="number">1</span><span class="special">)</span>
|
||
<span class="identifier">CardinalCubicHermiteAOS</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span><span class="identifier">_RMS</span> <span class="number">11</span> <span class="special">%</span>
|
||
</pre>
|
||
<p>
|
||
The logarithmic complexity of the non-equispaced version is evident, as is
|
||
the better cache utilization of the "array of structs" version as
|
||
the problem size gets larger.
|
||
</p>
|
||
</div>
|
||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||
<td align="left"></td>
|
||
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||
Walker and Xiaogang Zhang<p>
|
||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||
</p>
|
||
</div></td>
|
||
</tr></table>
|
||
<hr>
|
||
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